Low-density parity-check (LDPC) codes have been often used for error control codes in digital data communications including radio and optical communications networks because these codes achieve close to theoretical Shannon limit in practice. For those communications networks, LDPC-coded data are sent from a transmitter to a receiver over noisy channels. Using LDPC codes with an appropriate code rate, potential errors caused by channel noise can be efficiently corrected by an LDPC decoder at the receiver. To achieve better performance depending on the channel quality, an adaptive modulation and coding (AMC) has been used. The AMC uses different pairs of modulation order and code rate depending on the channel quality. For example, depending on a signal-to-noise ratio (SNR) at the receiver, the transmitter changes the modulation format, e.g., from 4-ary quadrature-amplitude modulation (QAM) to 16-ary QAM, and the code rate of the LDPC codes, e.g., from 0.5 to 0.9. By adaptively choosing the modulation order and the code rate, the AMC can achieve the maximum possible data rate close to Shannon limit over all the SNR regimes.
Typically, a degree distribution of LDPC codes determines a bit-error rate (BER) performance of the LDPC decoder. However, one optimized degree distribution of irregular LDPC code at a certain condition is not always best for other conditions even when the channel quality is constant. For example, the best code for 4 QAM may be no longer the best for 16 QAM. Therefore, the conventional AMC with the rate adaptation does not resolve this problem because the AMC does not provide multiple LDPC codes having the same code rate.
Recently, high-dimensional modulation (HDM) formats, such as polarization-switched quadrature phase-shift keying (PS-QPSK) and set-partitioned 128 QAM, have been used for reliable data transmissions, especially for coherent optical communications. Those modulation formats can provide a larger squared Euclidean distance, and the BER performance can be improved especially in uncoded networks. However, there is no obvious way to optimize LDPC codes for high-order and high-dimensional modulations. For example, one LDPC code optimized for 4-dimensional modulation format cannot be optimal for 8-dimensional modulation formats.
Some networks use bit-interleaved coded-modulations (BICM) and its iterative demodulation (ID) variant (called BICM-ID) for high-order modulation formats. Another way includes multi-layer coding (MLC), which uses multiple LDPC codes for different significant bits in high-order modulation formats. BICM is the simplest and optimizing LDPC codes for BICM does not depend on modulation formats, while the performance depends on labeling. BICM-ID outperforms BICM, and approaches the MLC performance bound. However, BICM-ID requires a higher latency because soft-decision information needs to be fed back to a demodulator from the decoder. Although MLC performs the best in theory, it has a drawback of codeword length shortening for each layered codes. In addition, there is no good way to design LDPC codes for HDM formats.
The performance of LDPC codes can be analyzed by an extrinsic information transfer (EXIT) chart or density evolution (DE). Those methods are also used for designing a degree distribution of irregular LDPC codes, for example, by linear programing for curve fitting. Although a good degree distribution can be designed with EXIT or DE, those methods assume infinite codeword length, infinite precision, and infinite number of iterations for decoding. Therefore, sometimes the expected performance cannot be obtained in practical use, where there exist some limitations in size of memory, bit width for precision, and maximum number of iterations.
Accordingly, there is a need in the art for an approach in designing practical LDPC codes to support various high-order and high-dimensional modulation formats in high-speed communications networks accounting for the trade-off between performance and complexity.